A. Sucker Rod Pump System
Reciprocating pump systems, such as sucker rod pump systems, extract fluids from a well and employ a downhole pump connected to a driving source at the surface. A rod string connects the surface driving force to the downhole pump in the wall. When operated, the driving source cyclically raises and lowers the downhole pump, and with each stroke, the downhole pump lifts well fluids toward the surface.
For example, FIG. 1 shows a sucker rod pump system 10 used to produce fluid from a well. A downhole pump 14 has a barrel 16 with a standing valve 24 located at the bottom. The standing valve 24 allows fluid to enter from the wellbore, but does not allow the fluid to leave. Inside the pump barrel 16, a plunger 20 has a traveling valve 22 located at the top. The traveling valve 22 allows fluid to move from below the plunger 20 to the production tubing 18 above, but does not allow fluid to return from the tubing 18 to the pump barrel 16 below the plunger 20. A driving source (e.g., a pump jack 11) at the surface connects by a rod string 12 to the plunger 20 and moves the plunger 20 up and down cyclically in upstrokes and downstrokes.
During the upstroke, the traveling valve 22 is closed, and any fluid above the plunger 20 in the production tubing 18 is lifted towards the surface. Meanwhile, the standing valve 24 opens and allows fluid to enter the pump barrel 16 from the wellbore. The highest point of the plunger's motion is typically referred to as the “top of stroke” (TOS), while the lowest point of the pump plunger's motion is typically referred to as the “bottom of stroke” (BOS).
At the TOS, the standing valve 24 closes and holds in the fluid that has entered the pump barrel 16. Additionally, at the TOS, the weight of the fluid in the production tubing 18 is supported by the traveling valve 22 in the plunger 20 and, therefore, also by the rod string 12, which causes the rod string 12 to stretch.
During the downstroke, the traveling valve 22 initially remains closed until the plunger 20 reaches the surface of the fluid in the barrel 16. Sufficient pressure builds up in the fluid below the traveling valve 22 to balance the pressure. The build-up of pressure in the pump barrel 16 reduces the load on the rod string 12 so that the rod string 12 relaxes.
This process takes place during a finite amount of time when the plunger 20 rests on the fluid, and the pump jack 11 at the surface allows the top of the rod string 12 to move downward. The position of the pump plunger 20 at this time is known as the “transfer point” because the load of the fluid column in the production tubing 18 is transferred from the traveling valve 22 to the standing valve 24. This results in a rapid decrease in load on the rod string 12 during the transfer.
After the pressure balances, the traveling valve 22 opens and the plunger 20 continues to move downward to its lowest position (i.e., the BOS). The movement of the plunger 20 from the transfer point to the BOS is known as the “fluid stroke” and is a measure of the amount of fluid lifted by the pump 14 on each stroke. In other words, the portion of the pump stroke below the transfer point may be interpreted as the percentage of the pump stroke containing fluid, and this percentage corresponds to the pump's fillage. Thus, the transfer point can be computed using a pump fillage calculation.
If there is sufficient fluid in the wellbore, the pump barrel 16 may be completely filled during an upstroke. Yet, under some conditions, the pump 14 may not be completely filled with fluid on the upstroke so there may be a void left between the fluid and the plunger 20 as it continues to rise. Operating the pump system 10 with only a partially filled pump barrel 16 is inefficient and, therefore, undesirable. In this instance, the well is said to be “pumped off,” and the condition is known as “pounding,” which can damage various components of the pump system. For a pumped off well, the transfer point most likely occurs after the TOS of the plunger 20.
Typically, there are no sensors to measure conditions at the downhole pump 14, which may be located thousands of feet underground. Instead, numerical methods are used calculate the position of the pump plunger 20 and the load acting on the plunger 20 from measurements of the position and load for the rod string 12 at the pump jack 11 located at the surface. These measurements are typically made at the top of the polished rod 28, which is a portion of the rod string 12 passing through a stuffing box 13 at the wellhead. A pump controller 26 is used for monitoring and controlling the pump system 10.
To efficiently control the reciprocating pump system 10 and avoid costly maintenance, a rod pump controller 26 can gather system data and adjust operating parameters of the system 10 accordingly. Typically, the rod pump controller 26 gathers system data such as load and rod string displacement by measuring these properties at the surface. While these surface-measured data provide useful diagnostic information, they may not provide an accurate representation of the same properties observed downhole at the pump. Because these downhole properties cannot be easily measured directly, they are typically calculated from the surface-measured properties.
Methods for determining the operational characteristics of the downhole pump 20 have used the shape of the graphical representation of the downhole data to compute various details. For example, U.S. Pat. No. 5,252,031 to Gibbs, entitled “Monitoring and Pump-Off Control with Downhole Pump Cards,” teaches a method for monitoring a rod pumped well to detect various pump problems by utilizing measurements made at the surface to generate a downhole pump card. The graphically represented downhole pump card may then be used to detect the various pump problems and control the pumping unit. Other techniques for determining operational characteristics are disclosed in U.S. Patent Publication Nos. 2011/0091332 and 2011/0091335, which are both incorporated herein by reference in their entireties.
B. Everitt-Jennings Method
In techniques to determine operational characteristics of a sucker rod pump system 10 as noted above, software analysis computes downhole data (i.e., a pump card) using position and load data measured at the surface. The most accurate and popular of these methods is to compute the downhole card from the surface data by solving a one-dimensional damped wave equation, which uses surface position and load as recorded at the surface.
Various algorithms exist for solving the wave equation. Snyder solved the wave equation using a method of characteristics. See Snyder, W. E., “A Method for Computing Down-Hole Forces and Displacements in Oil Wells Pumped With Sucker Rods,” Paper 851-37-K, 1963. Gibbs employed separation of variables and Fourier series in what can be termed the “Gibb's method.” See Gibbs, S. G. et al., “Computer Diagnosis of Down-Hole Conditions in Sucker Rod Pumping Wells,” JPT (Jan. 1996) 91-98; Trans., AIME, 237; Gibbs, S. G., “A Review of Methods for Design and Analysis of Rod Pumping Installations,” SPE 9980, 1982; and U.S. Pat. No. 3,343,409.
In 1969, Knapp introduced finite differences to solve the wave equation. See Knapp, R. M., “A Dynamic Investigation of Sucker-Rod Pumping,” MS thesis, U. of Kansas, Topeka (Jan. 1969). This is also the method used by Everitt and Jennings. See Everitt, T. A. and Jennings, J. W., “An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps,” SPE 18189, 1992; and Pons-Ehimeakhe, V., “Modified Everitt-Jennings Algorithm With Dual Iteration on the Damping Factors,” 2012 South Western Petroleum Short Course. The Everitt-Jennings method has also been implemented and modified by Weatherford International. See Ehimeakhe, V., “Comparative Study of Downhole Cards Using Modified Everitt-Jennings Method and Gibbs Method,” Southwestern Petroleum Short Course 2010.
To solve the one-dimensional wave equation, the Everitt-Jennings method uses finite differences. The rod string is divided into Mfinite difference nodes of length Li (ft), density ρi (lbm/ft3) and area Ai (in2). If we let u=u(x, t) be the displacement of position x at time t in a sucker rod pump system, the condensed one-dimensional wave equation reads:
                                          v            2                    ⁢                                                    ∂                2                            ⁢              u                                      ∂                              x                2                                                    =                                                            ∂                2                            ⁢              u                                      ∂                              t                2                                              +                      D            ⁢                                          ∂                u                                            ∂                t                                                                        (        1        )            
where the acoustic velocity is given by:
  v  =                    144        ⁢        Eg            ρ      and D represents a damping factor.
The first and second derivatives with respect to time are replaced by the first-order-correct forward differences and second-order-correct central differences. The second derivative with respect to position is replaced by a slightly rearranged second-order-correct central difference.
In the method, the damping factor D is automatically selected by using an iteration on the system's net stroke (NS) and the damping factor D. The damping factor D can be computed by the equation:
                    D        =                                                            (                550                )                            ⁢                              (                                  144                  ⁢                  g                                )                                                                    2                            ⁢              π                                ⁢                                                    (                                                      H                    PR                                    -                                      H                    H                                                  )                            ⁢                              τ                2                                                                    (                                                      ∑                                                                                                  ⁢                                                            ρ                      i                                        ⁢                                          A                      i                                        ⁢                                          L                      i                                                                      )                            ⁢                              S                2                                                                        (        2        )            
Where HPR is the polished rod horsepower (hp), S is the net stroke (in), τ is the period of one stroke (sec.), and HHYD is the hydraulic horsepower (hp) obtained as follows:HHYD=(7.36·10−6)QγFl  (3)where Q is the pump production rate (B/D), γ is the fluid specific gravity, and Fl is the fluid level (ft). The pump production rate is given by:Q=(0.1166)(SPM)Sd2  (4)where SPM is the speed of the pumping unit in strokes/minute, and d is the diameter of the plunger.
Additional details on the derivation of the damping factor D in equation (2) and the original iteration on the net stroke and damping factor algorithm are provided in Everitt, T. A. and Jennings, J. W., “An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps,” SPE 18189, 1992.
A modified Everitt-Jennings method also uses finite differences to solve the wave equation. As before, the rod string is discretized into M finite difference elements, and position and load (including stress) are computed at each increment down the wellbore. Then, as shown in FIG. 2, an iteration is performed on the net stroke and damping factor, which automatically selects a damping factor for each stroke.
The wave equation is initially solved to calculate the downhole card using surface measurements and an initial damping factor D set to 0.5 (Block 41). The initial net stroke S0 is determined from the computed card, and the fluid level in the well is calculated (Block 44 42). At this point, a new damping factor D is calculated from equation (2) (Block 43) and so forth, and the downhole card is again computed with the new damping factor D (Block 44). Based on the recalculated downhole card, a new net stroke S is determined (Block 45).
At this point, a check is then made to determine whether the newly determined net stroke S is close within some tolerance C of the initial or previous net stroke (Decision 46). If not, then another iteration is needed, and the process 40 returns to calculating the damping factor D (Block 43). If the newly determined net stroke is close to the previously determined net stroke (yes at Decision 46), then the iteration for determining the net stroke can stop, and the process 40 continues on to iterate on the damping factor D using the converged net stroke S (Block 47). The downhole data is then calculated using the newly calculated damping factor D (Block 48), and the pump horsepower Hpump is then calculated (Block 49a).
At this point, a check is made to see if the pump horsepower Hpump is close within some tolerance to the hydraulic horsepower Hhyd (Decision 49b). If so, then the process 40 ends as successfully calculating the downhole pump card with converged net stroke and damping factor D (Block 49c). If the pump horsepower Hpump and the hydraulic horsepower Hhyd are not close enough (no at Decision 49b), then the process 40 adjusts the current damping factor D by a ratio of the pump horsepower Hpump and the hydraulic horsepower HHyd (Block 49d). The process 40 of calculating the pump card with this adjusted damping factor D is repeated until the values for the pump and hydraulic horsepower Hpump and HHyd are close within the specified tolerance (Blocks 18 through 49d).
The advantage of the automatic iteration on the net stroke and the damping factor D as set forth above is that the damping factor D is adjusted automatically without human intervention. Thus, users managing a medium group to a large group of wells do not have to spend time manually adjusting the damping factor D as may be required by other methods.
C. Downhole Card and Damping
FIG. 3A shows exemplary surface data 50 obtained at the surface of the well. Load (y-axis) is graphed in relation to position (x-axis) as measured at the surface by a dynamometer system or the like. Using the techniques discussed previously, the measured surface data 50 can be mathematically translated to downhole data or pump card 60, which is shown ideally in this figure.
The pump card 60 has an upstroke fluid load line 62 (F0up) and a downstroke fluid load line (F0down). The height 63 of the pump card 60 is referred to as the fluid stroke F0, where F0=the upstroke fluid load line 62 (F0up)—the downstroke fluid load line 64 (F0down).
The pump or downhole stroke (PS) refers to the measure of extreme travel of the rod derived at the location of the pump. Thus, the “pump stroke” refers to the maximum displacement minus the minimum displacement and corresponds to the horizontal span or width of the downhole pump card 60.
Yet, the net stroke 68 (NS) refers to the measure of the portion of the pump stroke (PS) during which the fluid load is supported by the pump's standing valve. For a pumped off card 60′ as shown in FIG. 3B, the net stroke 68 (NS) is measured relative to the transfer point 66, which is the displacement in the pump stroke where load is transferred from the pump's traveling valve to the standing valve. (The transfer point can be computed using a pump fillage calculation.) The transfer point 66 occurs because the pressure in the pump barrel has exceeded the pressure in the plunger. The portion of the stroke below (with lower displacement than) the transfer point 66 is the net stroke NS and is interpreted as the portion of the pump stroke (PS) that actually contains liquid.
The displacement and load data can be used to determine one or more characteristics of the downhole pump's operation, such as the minimum pump stroke, the maximum pump stroke, and the transfer point in the downhole stroke. In turn, the area A of the pump card 60 or 60′ gives the pump horsepower of the downhole pump (20).
Using the wave equation as noted previously, the downhole pump card 60 is calculated from the surface data 50. The calculation requires that a damping factor D be used in the wave equation to add or remove energy from the calculation. If the calculation is over-damped as shown in FIG. 3C, then the downhole card 60A will be calculated with a shape as schematically shown. By contrast, if the calculation is under-damped, then the downhole card 60B will be calculated with a shape as schematically shown in FIG. 3D.
When analyzing surface data 50 and calculating the downhole card 60, the fluid load lines 62 and 64 represent the maximum and minimum loads applied to the rod string (12) by the pump (20) based on the current fluid level. When gas measurements are available, the fluid load lines 62 and 64 can be readily computed using the pump's intake pressure and the pump's discharge pressure. In the absence of these measurements, however, the fluid load lines 62 and 64 must be computed by other means.
In general, the fluid load lines 62 and 64 can be drawn on a graphical representation of the downhole card 60 because the fluid load lines 62 and 64 can generally be identified visually. When dealing with a large group of wells, however, any type of visual determination of fluid load lines 62 and 64 is highly impractical. Thus, because the fluid load lines 62 and 64 determine the fluid load used to compute the volumetric displacement of the pump (20) as well as fluid levels of the well, being able to determine fluid load lines 62 and 64 based on measured and calculated data can be quite useful for operating and diagnosing sucker rod pump systems.